#72222930

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁴×C₃³ ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃×(C₃⋊D₃)⋊A₄
commutator, cosocle	C₂³×C₃³	C₂³×C₃
Frattini	C₂×C₃	C₂×D₃²×A₄
Fitting	C₂⁴×C₃³	C₂²×C₃
socle	C₂³×C₃²	C₂²×C₃×D₃
hypercenter	C₂²	D₃×(C₃⋊D₃)⋊A₄
hypocenter	C₂²×C₃³	C₂×C₃×D₄
LCS	G ▷ C₂³×C₃³ ▷ C₂²×C₃³
UCS	C₁ ◁ C₂ ◁ C₂²

4520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^3,3^3|c^a=ce,f^d=fg,g^d=f,h^b=h^2,h^d=hj,i^a=i^2,i^b=i^2,j^b=j^2

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222936
maximal normal subgroup of	#72222993

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 3, 3, 3]);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.6);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);